What is the smallest number that can be written as the difference of 2 squares in 4 ways?

Differences of squares that are an even number apart are even, and those that are an odd number apart are odd. Since the smallest difference is to be 1 apart, the odds will be smaller. We need a number, X that can be written as:

(n+1)^2 - n^2 = 2n + 1
(n+3)^2 - n^2 = 6n + 9 = 3(2n+3) = 3*(odd number)
(n+5)^2 - n^2 = 10n + 25 = 5(2n+5) = 5*(odd number)
(n +7)^2 - n^2 = 14n + 49 = 7(2n+7) = 7*(odd number)

So our number must be odd, and must be a multiple of 3, 5, and 7. Well, 3*5*7 = 105 meets these criteria. Let's check by computing the various squares.

105 = 14n + 49 when n = 4
      = 10n + 25 when n = 8
      = 6n + 9 when n = 16
      = 2n + 1 when n = 52

So the differences are:
53^2 - 52^2  = 105*1
19^2 - 16^2 = 35*3 = 105
13^2 - 8^2 = 21*5 = 105
11^2 - 4^2 = 15*7 = 105

Posted by Paul on 2014-06-04 18:08:56